It might appear silly but I had this question always...

Why is the interval of 2 semitones called 'Major 2nd'? This interval occurs both in Major and Minor Scales. So, shouldn't it be called a Perfect 2nd?

All the other intervals obey this rule:

If they occur in Major scale, they are named "Major 3rd or 6th or 7th"
If they occur in Minor scale, they are named "Minor 3rd or 6th or 7th"
If they occur in both scales, they are named "Perfect unison or 4th or 5th or octave"...

If an interval of 2 semitones also follows this rule, it should be called Perfect 2nd and the interval of 1 semitone should be called Diminished 2nd and not Minor 2nd.

Or is there any other reason for their names? I mean....is it because unison, 4th, 5th and octave are used in perfect (or nearly perfect) cadences, they are called "perfect"?

Because it exists as
minor and major second (as well as diminished and augmented) unlike a 4th and 5th which can only be perfect, diminished or augmented.

C-D maj 2nd
E-F min 2nd

The reason is to do with ratios of their wavelengths.
Unison is obviously 1:1
An octave is 1:2 (The left hand side being the lower note)
A perfect 5th is nearly 2:3
A perfect 4th is nearly 3:4

A major 2nd isn't perfect because it isn't a 'clean' inteval. If you compare it with a perfect one, the perfect one will sound 'cleaner'.

Besides, nothing's ever perfect in life. That's what makes you human.

Apart from intervals. They're not human, so they're allowed to be perfect.

Can't there by doubly augmented/doubly diminished 4ths and 5ths?

The only perfects are the 4th and 5th. These are either perfect, augmented, or diminished. 2nd, 3rd, 6th and 7th are major or minor.

Ya, I agree that a major 2nd does not sound as clean as perfect 4th or 5th. One interesting thing that I observed is...

For an interval to sound 'clean', it should be
1) zero (unison) or
2) exactly an octave higher or
3) as far as possible from the lower note in the current octave as well the same note in the above octave...meaning it should be 4th or 5th!

But I didn't really understand what you are saying in ratios! Every interval can be expressed as some or the other ratio right? What's special about 2:3 and 3:4?

In extremes of chromatic writing you could probably define these intervals but in diatonic writing, no. You won't find them up to grade 8. They'd have no meaning in diatonic (keys) except perhaps rarely as passing or aux notes.

As far as I know....but............there'll be someone who can quote something...

No comment. By all means be intrigued but best not to get too cluttered with frequency ratios and such.
A 2nd which is what you asked about can exist in diatonic harmony as
maj 2nd
min 2nd
diminished second
augmented second
a diminished second (on a piano) sounds like a unison.

But...it's time to go home....

So, for example C - D = major 2nd
C - D# = augmented 2nd
C - Db = minor 2nd
C - Dbb = diminished 2nd.

This really needs a diagram....

Diagram is from http://www.miqel.com/spirit-jazz/vibrational-truth.html



N=1 is the fundamental, the note the you play. When you play another note with (or against) it, it will match up to another note in the harmonic series of your first note.

For reference, we'll say that

N=1 C
N=2 C up the octave
N=3 G
N=4 C up 2 octaves
N=5 E

If the diagram were to continue

N=6 G up the octave
N=7 a note between A & B-flat but closer to B-flat
N=8 C up 3 octaves
N=9 D
N=10 E up the octave

N=20 would indicate a speed of vibration that is 20 times faster than N=1, and twice as fast as N=10.

(note that the harmonics of a vibrating string are exactly the same as a vibrating air column or any other method of producing sound)

Now, I'll try not to get too complicated, let me know if something needs more explaining.

A "perfect" interval is called perfect because it sounds consonant (as opposed to dissonant, which a major 2nd interval is when the notes are played simultaneously).

Dissonance and consonance are governed by the inescapable harmonic series (see above diagram). Dissonance is created by two things vibrating at different speeds. The more different the speed, the more dissonant, Because the waves "clash" more often.

A unison interval has two things playing the same pitch -- "C" -- which will be vibrating at exactly the same speed. The is a "Perfect" interval (incidentally, the "most" Perfect).

An octave ("C up the octave") vibrates twice as fast as N=1. The vibrations only clash half the time, so it is considered to be pretty consonant -- and hence, "Perfect"

A fifth (the difference between "C up the octave" & "G") is also relatively consonant, it only clashes with "C" 2/3 of the time. Another Perfect interval.

A fourth (the difference between "G" and "C up 2 octaves) only clashes with "C" 3/4 of the time. This is the last Perfect interval, as anything past that is really pushing the limits of consonance, thirds (the next interval in the harmonic series) at one time were considered horribly dissonant.

To answer your specific question:

If you consult my notes at the top, you'll notice that "D" is an extremely long way away from "C", the vibrations clash 9/8 of the time! That's very dissonant, hence it is not a perfect interval.

I hope this helps you

It just occurred to me I should also explain the Major, minor, Augmented, diminished intervals, too. (mental note: read entire posts)

This has got to do with the way they are written on the staff.

C - E is a major third, right? If we make that interval bigger, it becomes C - E#, an augmented third

C - Eb is a minor third, and if we make it smaller is becomes C - Ebb, a diminished third.

So diminished = really small
minor = small
major = big
augmented = really big

The important thing to remember is that it's based on how it's written.

C - D is a major 2nd, but C - Ebb (enharmonic equivalent) is a diminished 3rd.
C - F# is an augmented 4th, but C - Gb is a diminished 5th

C - Cb : diminished unison
C - C : Perfect unison (it's a perfect interval because it's consonant, remember?)
C - C# : Augmented unison
C - Dbb : diminished 2nd
C - Db : minor 2nd
C - D : Major 2nd
C - D# : Augmented 2nd
C - Ebb : diminished 3rd
C - Eb : minor 3rd
C - E : Major 3rd
C - E# : Augmented 3rd

You get the picture

You didn't explain that all pitched instruments (and much else except an electronic tone generator of exceptional purity and (I think) a tuning fork) produce some overtones.
A clarinettist will spot a problem with your outlay of the harmonic series.

It just occurred to me I should also explain the Major, minor, Augmented, diminished intervals, too. (mental note: read entire posts)(quoting Zauberfaggott)

I think someones already done that. Do you read the preceding posts?


(As there's an edit button I edited part of my reply.)

I was about to post about this when I realised I'd done this before...



So that's the conclusion I came to when I asked the question. There is a thread about it here if you're interested.

Oh, sorry I didn't see the other post explaining that. I saw mainly the stuff about perfect intervals. It's zauberfagott, not zauberisha, I'm only human once again, sorry.

I didn't explain about overtones because I was trying not to go on for too long (in other forums I've been with longer I'm notorious for that), and I was trying not to be too complicated about it, otherwise I would have a post that was faaaaaar too long.

And I wanted to keep it simple rather than assume every reader is going have been to the same lectures, asked the same questions and read the same books that I have.

Besides, the overtones are almost invariably within the harmonic series, that's why playing a G followed by a C sounds logical even without the rest of the chord (diminished fifth between the 5th and 7th harmonics). The exception, of course, would be multiphonics (except double, triple etc. harmonics which are still in the harmonic series). You can hear that, anyway, if you listen carefully when you're playing a note on it's own.

So let me clarify...

The model of harmonics I used applies to vibrating strings (including vocal chords), wind instruments whose bored are either open cylinders (traversas) or closed cones (bassoons, oboes, saxes, etc.), and mallet percussion.

Clarinets have a different bore -- a closed cylinder -- so even numbered (correct me if I'm wrong) harmonics are difficult (but not impossible) to produce as they are very weak.

But any instrument has weaker or stronger harmonics.

Is that right?

My recollection is that later in that thread you realised that you had got your minors and majors mixed up when discussing 2nds and 7ths and corrected the paragraph. It should have read:

"A major second inverts a minor seventh which has ratio 16:9 and thus must itself have a ratio of 9:8; and a minor second must invert a major seventh which has ratio of 15:8 and thus must itself have a ratio of 16:15."

Actually some of the posts in this thread give a simplified version of musical acoustics which does not bear really close inspection. The integer values that have been quoted are the ideal ones. Integer values do occur in the analysis of repeating waveforms, such as those of bowed strings and blown wind instruments, but plucked and struck objects, like pianos and pizzicato strings usually behave slightly, or sometimes (e.g. bells and drums) very differently. Also, the precise tuning implied by integer ratios for the intervals is best approximated by professional a capella singers in early music. Orchestras are rarely as close and piano tuners don't even try, because pianos sound well enough in equal temperament, in which the frequency ratios are based not on integers but on the twelfth root of 2, and can then be played in any key.

To go back to the original question, I suspect that the adjectives perfect, major and minor got themselves attached to the intervals in the way they did because of medieval harmony: before about 1400, the only consonances were the unison, the octave, the fifth and the fourth. Thirds were considered imperfect consonances in the 15th C; the music of John Dunstable introduced them to the French and they were the defining characteristic of the contenance angloise. Minor triads were still avoided as final chords as late as the 18th C: hence the tierce de Picardie and final chords (e.g. Mozart Requiem, IIRC) with bare fifths.

Considered as a notational system, the intervals are open ended, and doubly augmented and diminished intervals do indeed occur. I was intrigued to find, in Elgar's "Sing Unto the Lord", a transition from one section to another in which the basses finished a phrase on Bbb and started the next on the F# below. Because this goes from a B down to an F, it is a fourth, and since Bb to F is a perfect fourth, Bbb to F# is a doubly diminished one. I have found a doubly augmented unison (also melodic) in the slow movement of the Brahms Horn Trio.

Don't take as incontrovertable truth the assertion that consonance results from notes whose frequencies are in ratios of small integers. That is only a first approximation that works reasonably well for simple timbres in the middle of the audible range. The detail of perceptual dissonance and consonance (whether an interval sounds rough or smooth) is very complicated and depends upon both the register and the detailed constitution of the notes (its constituent sine waves). One illustration of that is the usual advice in a harmony text book to space the notes of a chord further apart in the bass register than in the treble - there are good psycho-acoustical reasons for this. Another is that major thirds on pianos (but not on the trumpet stop of an electronic organ) sound reasonably satisfactory even though their frequency ratio is 1.26, not 1.25 (=5/4).

Thanks everyone for the detailed explanations! Guess I would take an hour to read through and assimilate the whole stuff!

Thanks a lot!

Interesting. I suppose the perceived jar in close intervals (2nds 3rds) among the lower frequencies, (the bass) arise from the slower beat given by the difference of the sum+difference frequencies, at extreme low being recognisable as a pulse.

?

That is indeed the Helmholtz theory of dissonance, which hs been much investigated and somewhat refined by perceptual psychologists over the last 50 years. The consensus among them now is that acoustic dissonance* results from beats between partials when the beat frequencies fall within a particular range. The frequency difference that maximises the roughness from pairs of pure sine waves varies with the register: above top C it corresponds to about 3/4 of a semitone, IIRC. Below middle C it is a constant. Bear in mind, however, that the ear perceives rather little of the fundamental of low frequencies: it identifies the pitch on the basis of higher partials at harmonic frequencies or close approximations to them, so double basses playing a major third low on their E strings sound less rough than sine waves of the same pitch and loudness would. Even so, every composer should avoid close chords on double basses unless he wants a special effect.

* The distinction is made between acoustic dissonance, which is inherent to the auditory system, and musical dissonance, which is learnt. Computers can generate continuous sounds with non-harmonic partials, and octaves or fifths between them can be made to sound very rough, but musicians will still call them "consonances" because they have been educated to do so. Consequently trained musicians are inappropriate subjects for the psycho-acoustic experiments that investigate these phenomena.